Question

Is empty relation reflexive and symmetric always? Give reasons for your answer.

Answer 1

As these are conditional statements if the antecedent is false the statements would be true. And as the relation is empty in both cases the antecedent is false hence the empty relation is symmetric and transitive.

As A is not empty, there exists some element $aϵA$. As R is empty, a R a does not hold, hence R is not reflexive.

An equivalence relation on a non-empty set can't be empty because it's reflexive. So, for any $aϵA$, you have $(a,a)ϵR$. Now there is some $aϵA$.
From a slightly different point of view, an equivalence relation on A always contains the identity relation
$\Delta A=\left\{\right(a,a):aϵA\}$ which is empty if and only if A is empty.
It's true that the empty relation is transitive and symmetric (also antisymmetric, by the way) on every set.